If you want to change your Login Name to something more fancy than "player1234567" or choose your own Avatar, it's easy.

From Small World 2, you can access directly to your profile on the Web site. Go to the Settings screen by tapping the gear icon in the main menu, then the Login Settings button and the "Edit my Profile" link:

This will take you to the Web site on the My Account page, being logged in with your account.
On the left side, you will see links to change your Profile (go there to change your Login Name), your Avatar, your password, etc.